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      SUBROUTINE <a name="DLAS2.1"></a><a href="dlas2.f.html#DLAS2.1">DLAS2</a>( F, G, H, SSMIN, SSMAX )
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  -- LAPACK auxiliary routine (version 3.1) --
</span><span class="comment">*</span><span class="comment">     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
</span><span class="comment">*</span><span class="comment">     November 2006
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     .. Scalar Arguments ..
</span>      DOUBLE PRECISION   F, G, H, SSMAX, SSMIN
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  Purpose
</span><span class="comment">*</span><span class="comment">  =======
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  <a name="DLAS2.14"></a><a href="dlas2.f.html#DLAS2.1">DLAS2</a>  computes the singular values of the 2-by-2 matrix
</span><span class="comment">*</span><span class="comment">     [  F   G  ]
</span><span class="comment">*</span><span class="comment">     [  0   H  ].
</span><span class="comment">*</span><span class="comment">  On return, SSMIN is the smaller singular value and SSMAX is the
</span><span class="comment">*</span><span class="comment">  larger singular value.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  Arguments
</span><span class="comment">*</span><span class="comment">  =========
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  F       (input) DOUBLE PRECISION
</span><span class="comment">*</span><span class="comment">          The (1,1) element of the 2-by-2 matrix.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  G       (input) DOUBLE PRECISION
</span><span class="comment">*</span><span class="comment">          The (1,2) element of the 2-by-2 matrix.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  H       (input) DOUBLE PRECISION
</span><span class="comment">*</span><span class="comment">          The (2,2) element of the 2-by-2 matrix.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  SSMIN   (output) DOUBLE PRECISION
</span><span class="comment">*</span><span class="comment">          The smaller singular value.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  SSMAX   (output) DOUBLE PRECISION
</span><span class="comment">*</span><span class="comment">          The larger singular value.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  Further Details
</span><span class="comment">*</span><span class="comment">  ===============
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  Barring over/underflow, all output quantities are correct to within
</span><span class="comment">*</span><span class="comment">  a few units in the last place (ulps), even in the absence of a guard
</span><span class="comment">*</span><span class="comment">  digit in addition/subtraction.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  In IEEE arithmetic, the code works correctly if one matrix element is
</span><span class="comment">*</span><span class="comment">  infinite.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  Overflow will not occur unless the largest singular value itself
</span><span class="comment">*</span><span class="comment">  overflows, or is within a few ulps of overflow. (On machines with
</span><span class="comment">*</span><span class="comment">  partial overflow, like the Cray, overflow may occur if the largest
</span><span class="comment">*</span><span class="comment">  singular value is within a factor of 2 of overflow.)
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  Underflow is harmless if underflow is gradual. Otherwise, results
</span><span class="comment">*</span><span class="comment">  may correspond to a matrix modified by perturbations of size near
</span><span class="comment">*</span><span class="comment">  the underflow threshold.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  ====================================================================
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     .. Parameters ..
</span>      DOUBLE PRECISION   ZERO
      PARAMETER          ( ZERO = 0.0D0 )
      DOUBLE PRECISION   ONE
      PARAMETER          ( ONE = 1.0D0 )
      DOUBLE PRECISION   TWO
      PARAMETER          ( TWO = 2.0D0 )
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">     .. Local Scalars ..
</span>      DOUBLE PRECISION   AS, AT, AU, C, FA, FHMN, FHMX, GA, HA
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">     .. Intrinsic Functions ..
</span>      INTRINSIC          ABS, MAX, MIN, SQRT
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">     .. Executable Statements ..
</span><span class="comment">*</span><span class="comment">
</span>      FA = ABS( F )
      GA = ABS( G )
      HA = ABS( H )
      FHMN = MIN( FA, HA )
      FHMX = MAX( FA, HA )
      IF( FHMN.EQ.ZERO ) THEN
         SSMIN = ZERO
         IF( FHMX.EQ.ZERO ) THEN
            SSMAX = GA
         ELSE
            SSMAX = MAX( FHMX, GA )*SQRT( ONE+
     $              ( MIN( FHMX, GA ) / MAX( FHMX, GA ) )**2 )
         END IF
      ELSE
         IF( GA.LT.FHMX ) THEN
            AS = ONE + FHMN / FHMX
            AT = ( FHMX-FHMN ) / FHMX
            AU = ( GA / FHMX )**2
            C = TWO / ( SQRT( AS*AS+AU )+SQRT( AT*AT+AU ) )
            SSMIN = FHMN*C
            SSMAX = FHMX / C
         ELSE
            AU = FHMX / GA
            IF( AU.EQ.ZERO ) THEN
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">              Avoid possible harmful underflow if exponent range
</span><span class="comment">*</span><span class="comment">              asymmetric (true SSMIN may not underflow even if
</span><span class="comment">*</span><span class="comment">              AU underflows)
</span><span class="comment">*</span><span class="comment">
</span>               SSMIN = ( FHMN*FHMX ) / GA
               SSMAX = GA
            ELSE
               AS = ONE + FHMN / FHMX
               AT = ( FHMX-FHMN ) / FHMX
               C = ONE / ( SQRT( ONE+( AS*AU )**2 )+
     $             SQRT( ONE+( AT*AU )**2 ) )
               SSMIN = ( FHMN*C )*AU
               SSMIN = SSMIN + SSMIN
               SSMAX = GA / ( C+C )
            END IF
         END IF
      END IF
      RETURN
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     End of <a name="DLAS2.119"></a><a href="dlas2.f.html#DLAS2.1">DLAS2</a>
</span><span class="comment">*</span><span class="comment">
</span>      END

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